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Let T be a contraction acting on the Hilbert space H such that
$\lim\sb{n}\Vert T\sp nh\Vert \ne 0$, for every $0\ne h\in H$.
Then $[x,y]=\lim\sb{n}<T\sp nx,T\sp ny>$ (x,y$\in H)$ defines
a new scalar product on H, and T acts as an isometry on the new
scalar product space (H,[.,.]). Let $\tilde H$ denote the completion
of (H,[.,.]), and let $\tilde T$ be the continuous extension
of T from (H,[.,.]) to $\tilde H.$ It is proved that if the isometry
$\tilde T$ is non-reductive, i.e. if $\tilde T$ contains a unilateral
shift, then T possesses a non-trivial invariant subspace. This
result partially answers the question of {\it R. Teodorescu}
and (INVALID INPUT)V. Vasyunin posed in [Linear and complex analysis.
Problem book (1984; Zbl 0545.30038)]. \par
Futhermore, relying
on a theorem of {\it H. Bercovici} and {\it K. Takahashi} [J.
Lond. Math. Soc., II. Ser. 32, 149-156 (1985; Zbl 0536.47009)],
it is shown that if in addition to the previous assumptions even
$\lim\sb{n}\Vert T\sp{*n}h\Vert \ne 0$ holds, for every $0\ne
h\in H$, then T is reflexive. This theorem extends former results
of {\it P. Y. Wu} [Proc. Am. Math. Soc. 77, 68-72 (1979; Zbl
0417.47001)] and the author.